Identification of time-varying Hammerstein systems from ensemble data

A NARMAX method for the identification of time-varying joint stiffness

Dynamic joint stiffness defines the dynamic relationship between the position of a joint and the torque acting about it and can be separated into intrinsic and reflex components. Under stationary conditions, these can be identified using a nonlinear parallel-cascade algorithm that models intrinsic stiffness and reflex stiffness as parallel pathways. Experimental results demonstrate that both intrinsic and reflex stiffness depend strongly on the operating point defined by mean joint position and the activation level. Consequently, both intrinsic and reflex stiffness will appear to be time-varying (TV) whenever the operating point changes, as during movement. This paper describes and validates a new method for identification of TV ankle stiffness. The method is based on the TV nonlinear autorregresive, moving average exogenous (NARMAX) model class. Simulation results demonstrated that the algorithm can accurately estimate the TV parameters of the ankle stiffness. We conclude that the algorithm is potentially a powerful new tool for the study of joint stiffness during TV conditions.

Identification of time-varying intrinsic and reflex joint stiffness

Dynamic joint stiffness defines the dynamic relationship between the position of a joint and the torque acting about it and can be separated into intrinsic and reflex components. Under stationary conditions, these can be identified using a nonlinear parallel-cascade algorithm that models intrinsic stiffness-a linear dynamic response to position-and reflex stiffness-a nonlinear dynamic response to velocity-as parallel pathways. Experiments using this method show that both intrinsic and reflex stiffness depend strongly on the operating point, defined by position and torque, likely because of some underlying nonlinear behavior not modeled by the parallel-cascade structure. Consequently, both intrinsic and reflex stiffness will appear to be time-varying whenever the operating point changes rapidly, as during movement. This paper describes and validates an extension of the parallel-cascade algorithm to time-varying conditions. It describes the ensemble method used to estimate time-varying intrinsic and reflex stiffness. Simulation results demonstrate that the algorithm can track rapid changes in joint stiffness accurately. Finally, the performance of the algorithm in the presence of noise is tested. We conclude that the new algorithm is a powerful new tool for the study of joint stiffness during functional tasks.

Performance evaluation of an algorithm for the identification of time-varying joint stiffness

Previously, we described a time-varying, parallel-cascade system identification algorithm that estimates intrinsic and reflex stiffness dynamics. It uses an iterative technique, in conjunction with established, time-varying, identification methods, to estimate the two pathways from ensembles of input and output realizations having the same time-varying behavior. This paper presents the results of a study that systematically evaluated the performance of the algorithm. Simulations were used to determine the algorithm's ability to track rapid changes in dynamic stiffness, and quantify its performance limits. There was close agreement between the simulated and estimated joint stiffness demonstrating that the algorithm estimates stiffness correctly even when it changes rapidly. However, the algorithm's ability to identify the reflex pathway was shown to depend on the relative contributions of the intrinsic and reflex pathways to the overall torque. As the intrinsic contribution to the output grew it became increasingly difficult to identify the reflex pathway accurately. The quality of the reflex identification greatly improved as the number of realizations in the data ensembles increased. More realizations were needed as the signal-to-noise ratio decreased and the relative contribution of the reflex pathway decreased. For good results, under typical time-varying experimental conditions, between 500 and 800 realizations are required.

Non-linear modeling of the steady-state disturbance term in isometric force

The analysis of isometric force may provide early detection of certain types of neuropathology such as Parkinson's disease. Our long term goal is to determine if there are detectable differences between model parameters of healthy and unhealthy individuals. In this study we evaluate nonlinear dynamic system models of isometric force using Hammerstien models. The experiments involved subjects exerting isometric force over a range from 5% to 95% of maximal voluntary contraction. The findings suggest that the Hammerstien models provide adequate fit, reducing the number of nonlinear difference equations of higher order that were determined in our earlier work.

Identification of time-varying intrinsic and reflex joint stiffness

We have developed a time-varying, parallel- cascade system identification algorithm to separate joint stiffness into intrinsic and reflex components at each point in time throughout rapid movements. The components are identified using an iterative algorithm in which intrinsic and reflex dynamics are identified using separate time-varying (TV) techniques based on ensemble methods. An ensemble of input-output records having the same TV behavior is acquired and used to identify the system dynamics as impulse response functions at time increments corresponding to the sampling interval. Simulation studies showed that the time-varying, parallel-cascade algorithm performed well under realistic conditions with 99.9% VAF between simulated and predicted torque. To evaluate the performance of the algorithm under realistic conditions we applied it to an ensemble of experimental data acquired under stationary conditions. Results demonstrated that the TV estimates converged to those of the established time-invariant algorithm and allowed us to determine how variance of the TV estimates varied with the number of realizations in the ensemble.

A bootstrap term selection method for the identification of time-varying nonlinear systems

Most physical systems are nonlinear and often time-varying. Constructing accurate models for nonlinear systems require specialized model structures that include their nonlinearities, whereas models of time-varying systems must include the time courses of the model's parameters. This contribution implements a technique in which the time dependence of the system's parameters are modeled by projecting them onto one or more expansion bases. However, unless an appropriate set of bases is chosen, it is likely that many of the basis functions used in the expansion will not contribute appreciably to the final model and may cause inaccuracy in the parameter estimates. Our study addresses the selection of a minimal number of parameters for optimization from a basis expansion of a system's time variations. The bootstrap method is used to select significant basis coefficients and hence basis functions in order to improve the overall accuracy of the model. The performance of the algorithm is demonstrated on simulated data from a system used to model the reflex contribution to joint stiffness.

Time-varying parallel-cascade system identification of ankle stiffness from ensemble data

Measurement of joint dynamic stiffness during time-varying conditions is crucial to understand the role of joint mechanics during movement. Stiffness can be separated into intrinsic and reflex components, and are modeled as linear dynamic and Hammerstein systems, respectively. Time-varying identification methods using ensemble data have been developed previously for both pathways and were tested separately on simulated data. In this study, these algorithms were integrated into the time-varying, parallel-cascade identification method. Ankle dynamics were modeled during a ramp input and simulated impulse response functions (IRFs) were generated. Gaussian white noise was low-pass filtered and was convolved with the simulated systems over 500 realizations. The ensemble data was used to evaluate the new identification technique. The mean variances accounted for (VAFs) between the true and identified IRFs for the intrinsic and reflex pathways were 99.9% and 97.7%, respectively, demonstrating the technique's strong ability to predict the system's dynamics.

On the identification of Hammerstein systems with time-varying parameters

A growing emphasis on the analysis of time-varying systems has intensified the need for simpler and more efficient identification methods for these systems. In this contribution, we examine the time-varying Hammerstein structure, comprising a memoryless nonlinearity with time-varying parameters followed by a time-varying linear filter. Two existing approaches for the identification of these systems, ensemble approaches and basis expansion methods, are combined as a single algorithm to give a much improved estimation tool. The proposed algorithm, applied to data from a simulation of a time-varying Hammerstein system, is used to construct models of the reflex contribution to joint stiffness and the results obtained are compared to those using the basis expansion method alone.